By Claude E. Shannon
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This booklet is split into elements. the 1st one is solely algebraic. Its target is the type of quadratic varieties over the sector of rational numbers (Hasse-Minkowski theorem). it really is completed in bankruptcy IV. the 1st 3 chapters comprise a few preliminaries: quadratic reciprocity legislation, p-adic fields, Hilbert symbols.
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35 2. , px y = px py (apart possibly from a set of points of probability zero). 3. Consider a generalized averaging operation of the following type: Z p0 y = with Z Z ; ax y dx = ; ax y px dx ; ax y dy = 1 ; ; ax y 0 : Then the entropy of the averaged distribution p0 y is equal to or greater than that of the original distribution px. 4. We have ; H x y = H x + Hx y = H y + Hy x and : Hx y H y 5. Let px be a one-dimensional distribution. The form of px giving a maximum entropy subject to the condition that the standard deviation of x be fixed at is Gaussian.
0. Since the average signal power P is less than or equal S N C W log 1 + P N W log 1+ S N : S N and are limited to band W and peak S the result will be proved. Consider the p p ensemble of functions of the following type. A series of t samples have the same value, either + S or , S, then the p next t samplesphave the same value, etc. The value for a series is chosen at random, probability 12 for + S and 12 for , S. ), the output is peak limited to S. Furthermore the average power is nearly S and can be made to approach this by taking t sufficiently large.
The set of all functions of time containing no frequencies over W cycles per second. 3. The set of all functions limited in band to W and in amplitude to A. 4. The set of all English speech signals as functions of time. 1 For example with the set, f t = sint + ; we may give a probability distribution for , P . The set then becomes an ensemble. Some further examples of ensembles of functions are: ; ;:::; n) with the probability of fk being pk . 1. A finite set of functions fk t (k = 1 2 2.